Not quite: According to the cosmological standard model, the visible universe will continue to grow (ie new galaxies will continue to come into view) - but only asymptotically, ie until a maximum size given by the comological event horizon is reached. However, the parts of the universe that aren't gravitationally bound to us will become fainter and fainter and increasingly resdhifted, and eventually, we'll be unable to detect other theoretically visible galaxies due to technological limitations.
There's a cosmological event horizon. Light emitted from within will reach us in finite time, light emitted from without won't. Similar to how a distant observer will never see on object falling into a (stationary) black hole cross the Schwarzschild horizon, we won't see galaxies crossing the cosmological horizon.
To me, it would seem that were it finite, there would be a point at which one would look back, and see the galaxy and clusters that compose the universe; forward would be an expanse of nothingness. But if this isn't the case, then how I can keep progressing forward (presumably forever, as I can't hit an edge) through space, encountering galaxy after galaxy, but it is still finite?
Unless this is like RPG games where the edges wrap.
Good question, but find me the edge of an idealized balloon. Where is the edge of a sphere? As to why you won’t come back, remember that spacetime is expanding faster an faster, and you can only travel below light speed. Mind you that’s just one possibility. The universe at large could be a lot of different things, but as humans were causally disconnected from anything beyond the shrinking observable universe. Shrinking from our perspective at least, because of the aforementioned expansion and speed limit.
edit: excuse my french ;)
Now what lies beyond the universe's boundary surface? Is such a surface even present?